We introduce a novel class of Bayesian mixtures for normal linear regression models which incorporates a further Gaussian random component for the distribution of the predictor variables. The proposed cluster-weighted model aims to encompass potential heterogeneity in the distribution of the response variable as well as in the multivariate distribution of the covariates for detecting signals relevant to the underlying latent structure. Of particular interest are potential signals originating from: (i) the linear predictor structures of the regression models and (ii) the covariance structures of the covariates. We model these two components using a lasso shrinkage prior for the regression coefficients and a graphical-lasso shrinkage prior for the covariance matrices. A fully Bayesian approach is followed for estimating the number of clusters, by treating the number of mixture components as random and implementing a trans-dimensional telescoping sampler. Alternative Bayesian approaches based on overfitting mixture models or using information criteria to select the number of components are also considered. The proposed method is compared against EM type implementation, mixtures of regressions and mixtures of experts. The method is illustrated using a set of simulation studies and a biomedical dataset.

翻译：本文提出了一类新颖的贝叶斯混合模型，用于正态线性回归分析，该模型在预测变量的分布中引入了一个附加的高斯随机分量。所提出的聚类加权模型旨在同时捕捉响应变量分布与协变量多元分布中潜在的异质性，从而检测与潜在结构相关的信号。特别值得关注的潜在信号来源包括：（i）回归模型的线性预测结构，以及（ii）协变量的协方差结构。我们采用套索收缩先验对回归系数进行建模，并使用图套索收缩先验对协方差矩阵进行建模。通过将混合成分数量视为随机变量，并实施跨维度伸缩采样器，采用完全贝叶斯方法估计聚类数量。同时考虑了基于过拟合混合模型或使用信息准则选择成分数量的替代贝叶斯方法。所提出的方法与EM类实现、回归混合模型及专家混合模型进行了比较。通过一组模拟研究和生物医学数据集对该方法进行了验证。